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Civil Engineering (CE) Exam  >  Civil Engineering (CE) Notes  >  Einsteins A and B Coefficients and Expression for Energy Density

Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE) PDF Download

Einsteins A and B Coefficients and Expression for energy density
Let E1 and E2 be the two energy states of the system of atoms. Let N1 and N2 be the number of atoms with energy states E1 and E2 respectively. Let Uν represent the energy density of incident photon having frequency (ν). Hence the system absorbs and emits energy through the following processes

(a)Stimulated Absorption:
In case of Stimulated absorption, the rate of absorption is given by number of absorptions per second per unit volume. Rate of absorption depends upon, (1) Number of atoms in the lower energy state. (N1)
(2) The energy density (Uν).
Hence, the Rate of absorption ∝ N1Uν and Rate of absorption = B12N1Uν, where B12 is proportionality constant called Einsteins coefficient of induced absorption.

(b)Spontaneous emission:
In case of Spontaneous emission, the rate of spontaneous emission is given by number of spontaneous emissions per second per unit volume. Rate of Spontaneous emission depends upon,
(1) Number of atoms in the higher energy state (E2). i.e. (N2)
Hence, the Rate of Spontaneous emission ∝ N2
Rate of Spontaneous emission = A21N2.
Where A21 is proportionality constant Called Einsteins coefficient of Spontaneous emission.

(c)Stimulated Emission:
In case of Stimulated Emission The rate of Stimulated emission is given by number of Stimulated emissions per second per unit volume. Rate of Stimulated Emission depends upon,
(1) Number of atoms in the higher energy state (E2). i.e. (N2)
(2) The energy density (Uν)
Hence, the Rate of Stimulated emission ∝ N2 Uν
Rate of Stimulated emission = B21 N2 Uν, where B21 is proportionality constant Called Einsteins coefficient of Stimulated Emission.

Under Thermal Equilibrium the total Energy of the System remains unchanged. Hence number of photons absorbed is equal to number of photons emitted. Rate of Induced Absorption = Rate of Spontaneous emission + Rate of Stimulated Emission. Therefore

Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

After simplification
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

But according to the Boltzman’s law of distribution the number of atoms in the higher energy state E2 at any temperature “T” is given by
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

Substituting for N1/N2 in equation (5.1)
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

According to Planck’s Law of Radiation. The equation for Energy density is given by
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

Comparing (5.2) and (5.3) we have
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

(i.e. Probability of Stimulated absorption is equal to Probability of Stimulated emission.) Hence A21 and B21 can be replaced by A and B and the expression for the Energy density is given by,
Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

Where h is Plancks constant, k is Boltzmann Constant, T is Absolute Temperature, and c is the velocity of light.

 

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FAQs on Einsteins A and B Coefficients and Expression for Energy Density - Civil Engineering (CE)

1. What are Einstein's A and B coefficients?
Ans. Einstein's A and B coefficients are terms used in the theory of stimulated emission and absorption of light. The A coefficient represents the rate at which atoms emit photons, while the B coefficient represents the rate at which atoms absorb photons.
2. How do Einstein's A and B coefficients relate to energy density?
Ans. Einstein's A and B coefficients are used to derive an expression for the energy density of radiation. The energy density is proportional to the product of the A coefficient, the B coefficient, and the photon density.
3. What is the expression for energy density using Einstein's A and B coefficients?
Ans. The expression for energy density using Einstein's A and B coefficients is given by: Energy density = (A coefficient * photon density) / (B coefficient * speed of light).
4. How do Einstein's A and B coefficients affect the behavior of light in a medium?
Ans. Einstein's A and B coefficients determine the rates of emission and absorption of light by atoms in a medium. The higher the values of these coefficients, the more significant the effect on the behavior of light. They play a crucial role in phenomena such as stimulated emission, which is the basis for the operation of lasers.
5. Are Einstein's A and B coefficients constant for all materials?
Ans. No, Einstein's A and B coefficients can vary for different materials. The values of these coefficients depend on the specific atomic and molecular properties of the material. Different materials can have different rates of emission and absorption of photons, leading to variations in the A and B coefficients.
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